Finite mixtures of multivariate skew t-distributions: Some recent and new results

Statistics and Computing

Volumn 24, Issue 2, 2014, Pages 181-202

  Lee, Sharon ^a   McLachlan, Geoffrey J ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

EM algorithm; Mixture models; Skew normal distributions; Skew t component distributions

Indexed keywords

EID: 84893974463     PISSN: 09603174     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11222-012-9362-4     Document Type: Article

References (55)

1. Akaike, H.: A new look at the statistical model identification. Autom. Control 19, 716-723 (1974).
2. Arellano-Valle, R., Bolfarine, H., Lachos, V.: Bayesian inference for skew-normal linear mixed models. J. Appl. Stat. 34(6), 663-682 (2007).
3. Arellano-Valle, R. B., Azzalini, A.: On the unification of families of skew-normal distributions. Scand. J. Stat. 33, 561-574 (2006).
4. Arellano-Valle, R. B., Genton, M. G.: On fundamental skew distributions. J. Multivar. Anal. 96, 93-116 (2005).
5. Arnold, B. C., Beaver, R. J.: Skewed multivariate models related to hidden truncation and/or selective reporting. Test 11, 7-54 (2002).
6. Azzalini, A.: A class of distributions which includes the normal ones. Scand. J. Stat. 12, 171-178 (1985).
7. Azzalini, A.: The skew-normal distribution and related multivariate families. Scand. J. Stat. 32, 159-188 (2005).
8. Azzalini, A., Capitanio, A.: Distribution generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. R. Stat. Soc., Ser. B 65, 367-389 (2003).
9. Azzalini, A., Dalla, Valle A.: The multivariate skew-normal distribution. Biometrika 83(4), 715-726 (1996).
10. Banfield, J. D., Raftery, A.: Model-based gaussian and non-gaussian clustering. Biometrics 49, 803-821 (1993).
11. Basso, R. M., Lachos, V. H., Cabral, C. R. B., Ghosh, P.: Robust mixture modeling based on scale mixtures of skew-normal distributions. Comput. Stat. Data Anal. 54, 2926-2941 (2010).
12. Böhning, D.: Computer-Assisted Analysis of Mixtures and Applications: Meta-Analysis, Discase Mapping and Others. Chapman and Hall, New York (1999).
13. Branco, M. D., Dey, D. K.: A general class of multivariate skew-elliptical distributions. J. Multivar. Anal. 79, 99-113 (2001).
14. Brinkman, R., Gaspareto, M., Lee, S. J., Ribickas, A., Perkins, J., Janssen, W., Smiley, R., Smith, C.: High content flow cytometry and temporal data analysis for defining a cellular signature of graft versus host disease. Biol. Blood Marrow Transplant. 13, 691-700 (2007).
15. Cabral, C., Bolfarine, H., Pereira, J.: Bayesian density estimation using skew student-t-normal mixtures. Comput. Stat. Data Anal. 52, 5075-5090 (2008).
16. Cabral, C., Lachos, V., Prates, M.: Multivariate mixture modeling using skew-normal independent distributions. Comput. Stat. Data Anal. 56, 126-142 (2012).
17. Dempster, A., Laird, N. M., Rubin, D.: Maximum likelihood from incomplete data via the em algorithm. J. R. Stat. Soc., Ser. B 39, 1-38 (1977).
18. Everitt, B. S., Hand, D. J.: Finite Mixture Distributions. Chapman and Hall, London (1981).
19. Fraley, C., Raftery, A. E.: How many clusters? Which clustering methods? Answers via model-based cluster analysis. Comput. J. 41, 578-588 (1999).
20. Frühwirth-Schnatter, S.: Finite Mixture and Markov Switching Models. Springer, New York (2006).
21. Frühwirth-Schnatter, S., Pyne, S.: Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions. Biostatistics 11, 317-336 (2010).
22. Genz, A., Bretz, F.: Methods for the computation of multivariate t-probabilities. J. Comput. Graph. Stat. 11, 950-971 (2002).
23. Gómez, H., Venegas, O., Bolfarine, H.: Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics 18, 395-407 (2007).
24. González-Farás, G., Domínguez-Molinz, J. A., Gupta, A. K.: Additive properties of skew normal random vectors. J. Stat. Plan. Inference 126, 521-534 (2004).
25. Green, P. J.: On use of the em algorithm for penalized likelihood estimation. J. R. Stat. Soc. B 52, 443-452 (1990).
26. Gupta, A. K.: Multivariate skew-t distribution. Statistics 37, 359-363 (2003).
27. Ho, H., Lin, T., Chen, H., Wang, W.: Some results on the truncated multivariate t distribution. J. Stat. Plan. Inference 142, 25-40 (2012a).
28. Ho, H., Pyne, S., Lin, T.: Maximum likelihood inference for mixtures of skew student-t-normal distributions through practical em-type algorithms. Stat. Comput. 22, 287-299 (2012b).
29. Karlis, D., Santourian, A.: Model-based clustering with non-elliptically contoured distributions. Stat. Comput. 19, 73-83 (2009).
30. Karlis, D., Xekalaki, E.: Choosing initial values for the em algorithm for finite mixtures. Comput. Stat. Data Anal. 41, 577-590 (2003).
31. Kotz, S., Nadarajah, S.: Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge (2004).
32. Lachos, V. H., Ghosh, P., Arellano-Valle, R. B.: Likelihood based inference for skew normal independent linear mixed models. Stat. Sin. 20, 303-322 (2010).
33. Lee, S., McLachlan, G.: On the fitting of mixtures of multivariate skew t-distributions via the em algorithm (2011). arXiv: 1109. 4706 [statME].
34. Lin, T. I.: Maximum likelihood estimation for multivariate skew-normal mixture models. J. Multivar. Anal. 100, 257-265 (2009).
35. Lin, T. I.: Robust mixture modeling using multivariate skew t distribution. Stat. Comput. 20, 343-356 (2010).
36. Lin, T. I., Lee, J. C., Hsieh, W. J.: Robust mixture modeling using the skew-t distribution. Stat. Comput. 17, 81-92 (2007a).
37. Lin, T. I., Lee, J. C., Yen, S. Y.: Finite mixture modelling using the skew normal distribution. Stat. Sin. 17, 909-927 (2007b).
38. Lindsay, B. G.: Mixture Models: Theory, Geometry, and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 5. Institute of Mathematical Statistics, Hayward (1995).
39. Liseo, B., Loperfido, N.: A bayesian interpretation of the multivariate skew-normal distribution. Stat. Probab. Lett. 61, 395-401 (2003).
40. Liu, C., Rubin, D.: The ecme algorithm: a simple extension of the em and ecm with faster monotone convergence. Biometrika 81, 633-648 (1994).
41. Maier, L. M., Anderson, D. E., De Jager, P. L., Wicker, L., Hafler, D. A.: Allelic variant in ctla4 alters t cell phosphorylation patterns. Proc. Natl. Acad. Sci. USA 104, 18607-18612 (2007).
42. McLachlan, G., Peel, D.: Robust cluster analysis via mixtures of multivariate t-distributions. In: Amin, A., Dori, D., Pudil, P., Freeman, H. (eds.) Lecture Notes in Computer Science, vol. 1451, pp. 658-666. Springer, Berlin (1998).
43. McLachlan, G. J., Basford, K. E.: Mixture Models: Inference and Applications. Dekker, New York (1988).
44. McLachlan, G. J., Peel, D.: Finite Mixture Models. Wiley Series in Probability and Statistics (2000).
45. O'Hagan, A.: Bayes estimation of a convex quadratic. Biometrika 60, 565-571 (1973).
46. O'Hagan, A.: Moments of the truncated multivariate-t distribution (1976). http://www. tonyohagan. co. uk/academic/pdf/trunc_multi_t. PDF.
47. O'Hagan, A., Murphy, T., Gormley, I.: Computational aspects of fitting mixture models via the expectation-maximization algorithm. Comput. Stat. Data Anal. 56, 3843-3864 (2012).
48. Peel, D., McLachlan, G.: Robust mixture modelling using the t distribution. Stat. Comput. 10, 339-348 (2000).
49. Pyne, S., Hu, X., Wang, K., Rossin, E., Lin, T. I., Maier, L. M., Baecher-Allan, C., McLachlan, G. J., Tamayo, P., Hafler, D. A., De Jager, P. L., Mesirow, J. P.: Automated high-dimensional flow cytometric data analysis. Proc. Natl. Acad. Sci. USA 106, 8519-8524 (2009).
50. Sahu, S., Dey, D., Branco, M.: A new class of multivariate skew distributions with applications to bayesian regression models. Can. J. Stat. 31, 129-150 (2003). Eratum: Can. J. Stat. 37, 301-302 (2009).
51. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461-464 (1978).
52. Titterington, D. M., Smith, A. F. M., Markov, U. E.: Statistical Analysis of Finite Mixture Distributions. Wiley, New York (1985).
53. Vrbik, I., McNicholas, P.: Analytic calculations for the em algorithm for multivariate skew t-mixture models. Stat. Probab. Lett. 82, 1169-1174 (2012).
54. Wang, K.: EMMIX-skew: EM algorithm for mixture of multivariate skew normal/t distributions (2009). http://www. maths. uq. edu. au/gjm/mix_soft/EMMIX-skew, R package version 1. 0-12.
55. Wang, K., Ng, S. K., McLachlan, G. J.: Multivariate skew t mixture models: applications: applications to fluorescence-activated cell sorting data. In: Shi, H., Zhang, Y., Botema, M., Lovell, B., Maoder, A. (eds.) DICTA 2009 (Conference of Digital Image Computing: Techniques and Applications, Melbourne), pp. 526-531. IEEE Comput. Soc., Los Alamitos (2009).